While the proof is provided in Appendix , we remark that the optimality of the exponential schedule can be directly inferred from the result of [1]}: it was shown that, given a finite time-horizon of length \(T\) , a mean-variance optimization results in a trajectory \(X_t = \frac{ \sinh ( \kappa (T-t) ) }{ \sinh (\kappa T ) } X_0\)  for some constant \(\kappa > 0\) .
As \(T \nearrow \infty \) , we can observe that the optimized trajectory converges to an exponential schedule \(X_t = e^{ -\kappa t } X_0\) .
